Abstract: The understanding and modeling of failure processes in solids is a central task in materials sciences. Mathematical models typically result in highly nonlinear, coupled systems of partial differential equations, where additional nonsmooth constraints, as for instance the unidirectionality of evolution processes or the impenetrability of the material, have to be taken into account.

This minisymposium intends to discuss recent advances in the mathematical treatment of failure phenomena, and brings together scientists from the fields of modeling, analysis, and numerics. Analytical methods and numerical strategies both for (quasi-)static and rate-dependent, non-smooth failure models will be presented.

MS-We-E-06-116:00--16:30Analysis of Crystalline DefectsOrtner, Christoph (Univ. of Warwick)Abstract: I will present a general framework for the analysis of material defects embedded into a homogeneous crystalline environment. A key result in this framework is a generic regularity estimate that gives qualitatively sharp bounds on the "defect core". I will then show examples how this framework can be employed (i) in the analysis of dislocation models, (ii) multi scale simulation, and (iii) to construct atomistic models of material failure.

MS-We-E-06-216:30--17:00Phase-field approach for quasi-static evolutions in fracture mechanicsNegri, Matteo (Univ. of Pavia)Abstract: We consider a couple of quasi-static evolutions obtained by sequences of time-discrete incremental minimization problems generated by a locally minimizing movement (w.r.t. H^1 and L^2 norm) and by the alternate minimization scheme. We characterize their time-continuous limits as parametrized BV-evolutions in terms of stationarity and energy balance. We provide then some physical properties in terms of energy release and thermodynamical consistency of the irreversibility constraint.

MS-We-E-06-317:00--17:30An irreversible gradient flow and its application to a crack propagation modelKimura, Masato (Kanazawa Univ.)Abstract: We consider a nonlinear diffusion equation with irreversible property and construct a unique strong solution by using implicit time discretization. A new regularity estimate for the obstacle problem is established and is used in the construction of the strong solution. An application to a crack propagation model is also presented.

MS-We-E-06-417:30--18:00Dynamics of microstructure: the example of a damage modelGarroni, Adriana (Sapienza, Univ. of Rome)Abstract: Many models in material science (as plasticity, damage, phase transition or fracture), involve non convex energies. This lack of convexity is responsible for the formation of microstructure and represents a serious issue in the study of evolution problems.
I will focus on a simple, but yet enough rich, model for elastic brittle damage introduced by Francfort and Marigo in which many of main questions related to evolution of microstructure can be successfully addressed.